Question

Use a graphing calculator to graph the original function y=1/2x−4
and the vertically dilated function with a factor k=6
. Determine the x-intercepts of both functions.(1 point)
Responses

The original function has an x-intercept of (8,0)
, and the dilated function has an x-intercept of (48,0)
.
The original function has an x -intercept of left parenthesis 8 comma 0 right parenthesis , and the dilated function has an x -intercept of left parenthesis 48 comma 0 right parenthesis .

The original function has an x-intercept of (8,0)
, and the dilated function has an x-intercept of (8,0)
.
The original function has an x -intercept of left parenthesis 8 comma 0 right parenthesis , and the dilated function has an x -intercept of left parenthesis 8 comma 0 right parenthesis .

The original function has an x-intercept of (0,−4)
, and the dilated function has an x-intercept of (0,−24)
.
The original function has an x -intercept of left parenthesis 0 comma negative 4 right parenthesis , and the dilated function has an x -intercept of left parenthesis 0 comma negative 24 right parenthesis .

The original function has an x-intercept of (−8,0)
, and the dilated function has an x-intercept of (−8,0)
.

Answer 1

To find the x-intercepts of the original function \( y = \frac{1}{2}x - 4 \) and its vertically dilated function with a factor \( k = 6 \), we first find the x-intercept of the original function.

### 1. Original Function
To find the x-intercept, set \( y = 0 \):

\[
0 = \frac{1}{2}x - 4
\]

Solving for \( x \):

\[
\frac{1}{2}x = 4 \\
x = 4 \times 2 \\
x = 8
\]

So, the x-intercept of the original function \( y = \frac{1}{2}x - 4 \) is \( (8, 0) \).

### 2. Vertically Dilated Function
The vertically dilated function with a factor \( k = 6 \) can be expressed as:

\[
y = 6 \left( \frac{1}{2}x - 4 \right) = 3x - 24
\]

To find the x-intercept of this dilated function, set \( y = 0 \):

\[
0 = 3x - 24
\]

Solving for \( x \):

\[
3x = 24 \\
x = \frac{24}{3} \\
x = 8
\]

So, the x-intercept of the dilated function \( y = 3x - 24 \) is also \( (8, 0) \).

### Conclusion
The x-intercepts of both functions are:

- Original function: \( (8, 0) \)
- Dilated function: \( (8, 0) \)

The correct response is:

**The original function has an x-intercept of (8,0), and the dilated function has an x-intercept of (8,0).**